What Derivative Rule does Trig Substitution match up with?

I was thinking about the Integral Rules and trying to understand why they work. It seems to me like all Integral Rules should have a Derivative Rule counterpart:

• U Substitution: Chain Rule. The chain rule says: $\frac d{dx}f(g(x)) = f'(g(x))\times g'(x)$ and U Sub removes a function and its derivative from the integral:$\int g'(x)f'(g(x))dx = \int f'(u)du$
• Integration by Parts: Product/Quotient Rule. Example: $\int x \arctan{x}dx$. Thinking using product rule, it can be seen that $\frac d{dx}\left(\frac12x^2 \arctan x\right)$ $=$ $\displaystyle x \arctan x + \frac x{2(1+x^2)}$. So $\int x \arctan x dx = \int \left(x \arctan x + \frac x{2(1+x^2)}\right)dx - \int \frac x{2(1+x^2)}dx = \frac12x^2\arctan x - \int \frac x{2(1+x^2)}dx$ which is the rule for Integration by Parts.
• Partial Fractions: Not really any derivative rule here; this is basically manipulating the function with algebra to make it easier to integrate.
• Trig Substitution: ?

What Derivative rule matches up with Trig Substitution? Why does Trig Substitution work?

It's easy to see how the first two methods were invented. How was Trig Substitution invented?

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Trig substitution is a special kind of u substitution, with u being a trigonometric function of some kind. – Cameron Williams Apr 29 '13 at 5:12
@CameronWilliams I sort of realized that. – Justin Apr 29 '13 at 5:31

$$\sin^2 + \cos^2 = 1\\ 1 + \cot^2 = \csc^2\\ \tan^2 + 1 = \sec^2$$
This can often make for simpler integrals (especially ones with ugly square roots), such as if we have a $\sqrt{a^2-x^2}$, we can substitute (among many possible substitutions) $x = a \cos \theta$, reducing the sqrt down to $a\sin\theta$. Basically, trig sub is a method of removing square roots. It might also be a method to do other things as well.